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May 4th, 2009, 01:02 PM   #1
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Finding the minimal polynomial

if f = x^3 + 2X+1 is the minimal polynomial of alpha, then what is the minimal polynomial of alpha^2?

I have no idea how to solve that. A hint pls.
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May 9th, 2009, 09:24 PM   #2
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Re: Finding the minimal polynomial

It's impossible, I think.
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May 9th, 2009, 11:33 PM   #3
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Re: Finding the minimal polynomial

Quote:
Originally Posted by lambda
It's impossible, I think.
Hardly. I'm making (thought not a very bold one) the assumption that this f(x) is a polynomial over Z, so this implies that the roots of f(x) are algebraic integers. It is a theorem (and not a trivial one at all!) that the algebraic integers form a ring and there is a method that is based on using eigenvalues of a matrix to compute the minimal polynomial of of ab, for some algebraic integers a and b. I'm a little rusty on this method and will have to look it over and figure it out, but impossible--definitely not.
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May 15th, 2009, 08:56 AM   #4
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Re: Finding the minimal polynomial

Thanks for the replies but I solved the problem myself by looking at the powers of alpha^2.

alpha^n can always be written as lin. combination of elements of the basis {0,alpha,alpha^2}

if I do so, then I see that a^6 is a linear combination of 1,a^2,a^4 and so by solving the resulting linear equations I found the minimal polynomial.
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